SOLUTION: Solve the following equation on the interval [0, 360 degrees]: sin2x=1 This is what I have done so far: Let 2x = theta sin theta = 1 theta = 90 degrees 2x = 90 --> x = 4

Question 1103336: Solve the following equation on the interval [0, 360 degrees]:
sin2x=1
This is what I have done so far:
Let 2x = theta
sin theta = 1
theta = 90 degrees
2x = 90 --> x = 45 degrees
Add period to theta:
90 + 180 = 270 --> x = 270/2 = 135 degrees
270+ 90 = 450 --> x = 450/2 = 225 degrees
I don't understand why the answer in my book says "x = 45 degrees, 225 degrees". According to my work done, shouldn't x = 135 degrees as well?
Help with this question would be really appreciated!
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52803) (Show Source): You can put this solution on YOUR website!
.

sin(2x) = 1 ====> 2x = . The last equation 2x = has TWO solutions in the interval [0,). One of these two solutions is = , which is OBVIOUS. The second solution is to add to the first solution: = = . Why ? - Because when you take with = , you will get = , which is GEOMETRICALLY THE SAME AS the ANGLE . So your is geometrically the same and satisfies the original equation sin(2x) = 1. Notice that both = and = belong to the interval [0,), so they both are the solutions to your original equation sin(2x) = 1 in this interval. The plot below confirms my arguments: Plot y = sin(2x) (red) and y = 1 (green)

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Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

Well, for one thing, 135 degrees does not satisfy the equation. ....

The period of sine is 360 degrees, not 180 degrees.

After you get your first answer of 2x=90, giving you the answer x=45 degrees, your next answer comes from 2x = 90+360 = 450, giving you the answer x=225 degrees.

I suspect that your confusion comes from thinking that, since the equation involves sin(2x), you only need to add 360/2=180 degrees to your first answer for 2x.

I always found it easier to avoid this kind of confusion by first solving the problem all the way through for the values of 2x and then getting my answers for x from those answers.

In this problem, we are to find the solutions to . Since our function is sin(2x), the prescribed interval becomes .

So now find all the solutions between 0 and 720 degrees where the value of sine is 1; they are at 90 degrees and 450 degrees. Then, since the values of 2x are 90 and 450 degrees, the values for x (the solutions we are to find) are 45 and 225 degrees.
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